I think what you are looking for is RetroPlug

Timing is derived from the crystal oscillator, the frequency of which is then divided to some much smaller value. So the CPU type doesn't really give you any advantage or disadvantage. I don't know where the master clock for LSDj comes from, but I'd have to assume it uses the screen refresh clock at ~60 Hz. I believe LSDj can be synced externally via MIDI, but I've never done that myself. Someone else will need to help you with that :]
What I'd do is I'd use the gameboy as the timing reference and sync the FL project with that, as it's easier to do.

As a bit of a side question, if you're willing to go the PSP emulation route (and that's what it is at that point, emulation), why not go the full emulation route on a PC then?

That being said, from what I understand the DMG is preferred for its sound because of its analog circuitry.

tl;dr DMG + either MIDI sync or sync your project to it instead. Just my two cents.

So I’m considering either the psp for emulation or an actual gameboy. I use synth tracks to tie in with my fl studio project but I’m looking to get genuine 8bit sounds while also adhering to the timing of my tracks.
I have experience with lsdj but not with putting the exported track into lsdj and using it in combination.

Would the timing be best on a gameboy since it’s official hardware? Or maybe the gameboy color since the cpu may have more headroom than the DMG?

Or would the psp be better since it’s much more powerful? I’m looking for best sound and timing here.
Thanks!